The real numbers

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The real numbers

[ilmath]\mathbb{R} [/ilmath]

The real line is the name given to the reals with their "usual topology", the topology that is induced by the absolute value metric

Definition

Cantor's construction of the real numbers

The set of real numbers, [ilmath]\mathbb{R} [/ilmath], is the quotient space, [ilmath]\mathscr{C}/\sim[/ilmath] where:

• [ilmath]\mathscr{C} [/ilmath] - the set of all Cauchy sequences in [ilmath]\mathbb{Q} [/ilmath] - the quotients
• [ilmath]\sim[/ilmath] - the usual equivalence of Cauchy sequences

We further claim:

1. that the familiar operations of addition, multiplication and division are well defined and
2. by associating [ilmath]x\in\mathcal{Q} [/ilmath] with the sequence [ilmath] ({ x_n })_{ n = 1 }^{ \infty }\subseteq \mathbb{Q} [/ilmath] where [ilmath]\forall n\in\mathbb{N}[x_n:=x][/ilmath] we can embed [ilmath]\mathbb{Q} [/ilmath] in [ilmath]\mathbb{R}:=\mathscr{C}/\sim[/ilmath]

Axiomatic construction of the real numbers

Axiomatic construction of the real numbers/Definition

[ilmath]\mathbb{R} [/ilmath] is an example of:

• Vector space
• Field ([ilmath]\implies\ \ldots\implies[/ilmath] ring)
• Complete metric space ([ilmath]\implies[/ilmath] topological space)
  • With the metric of absolute value

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TODO: Flesh out

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Properties

• The axiom of completeness - a badly named property that isn't really an axiom.

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If [ilmath]S\subseteq\mathbb{R} [/ilmath] is a non-empty set of real numbers that has an upper bound then:

• [ilmath]\text{Sup}(S)[/ilmath] (the supremum of [ilmath]S[/ilmath]) exists.